Machine learning quantum phases of matter beyond the fermion sign problem (1608.07848v1)

Abstract: State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green's function (but not the auxiliary field) holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green's function, e.g.~in the form of equal-time correlation functions, fail. We expect that this capacity of hierarchical machine learning techniques to circumvent the fermion sign problem will drive novel insights into some of the most fundamental problems in statistical physics.

• The paper demonstrates that convolutional neural networks reliably detect quantum phase transitions by analyzing Green's functions from auxiliary-field QMC simulations.
• The methodology circumvents the fermion sign problem by excluding auxiliary field data, enabling clear identification of critical transitions in complex fermion systems.
• Strong numerical evidence in Hubbard-type models confirms the approach's effectiveness in mapping phase boundaries in both spinful and spinless fermion systems.

Machine Learning Quantum Phases of Matter Beyond the Fermion Sign Problem

This paper introduces a novel approach of utilizing machine learning, specifically convolutional neural networks (CNNs), to classify quantum phases in many-fermion systems where traditional quantum Monte Carlo (QMC) techniques are hindered by the notorious fermion sign problem. The authors provide evidence of CNNs being able to correctly identify and locate quantum phase transitions by analyzing the Green's function obtained from auxiliary-field QMC simulations, even when the system suffers from severe sign problems.

The fermion sign problem emerges when sampling quantum mechanical systems due to the presence of positive and negative statistical weights, which interferes with typical probability interpretations. In swathes of quantum Hamiltonians, this complicates unbiased QMC simulations. The authors tackle this by leveraging machine learning to circumvent the sign problem, providing insights into fundamental statistical physics problems that traditional methods might fail to address.

The paper uses auxiliary-field QMC to sample quantum many-fermion systems to feed into CNNs. Notably, the CNN distinguishes between quantum phases using solely the Green's functions, excluding the auxiliary field. This claim challenges conventional approaches that utilize equal-time correlation functions, particularly in systems afflicted by significant sign problems.

Strong Numerical Results

The authors provide robust empirical evidence supporting their methodology across multiple models. They focus on the Hubbard-type quantum lattice models with competing itinerant and charge-ordered phases to showcase the CNN's ability to identify quantum phases:

1. Spinful Fermion Systems: The authors explore the quantum phase transition from a Dirac semi-metal to an antiferromagnetic insulator. The CNN accurately predicts a transition around the interaction parameter, aligning with known critical values, illustrating the effectiveness of their method even when visual differences in configurations are imperceptible.
2. Spinless Fermion Models: In examining sign-problematic models with no known basis to eliminate the issue, such as the spinless fermion models, the CNN succeeds in discernibly mapping the phase transition. This result is notable as it deviates from conventional sign-problematic QMC forecasts where statistical uncertainties typically obscure results.

Implications and Future Developments

The research represents a significant step towards integrating machine learning into quantum statistical physics, demonstrating the technique's utility in phase classification and critical point identification without being deterred by the sign problem. This hybrid QMC + machine learning framework not only enriches understanding of quantum phase transitions but also introduces a toolset for practical exploration in condensed matter physics, particularly in systems where sign problems preclude traditional QMC methods.

Future research directions could include extending this methodology to more complex quantum systems, investigating different neural network architectures, and potentially discovering a systematic link between machine learning frameworks and more traditional statistical physics concepts. Moreover, the paper opens avenues for improving computational efficiency in mapping phase diagrams and exploring systems that produce exotic phases unattainable through classical approaches.

Overall, this work underlines the increasing relevance of interdisciplinary techniques, such as the intersection of machine learning and quantum physics, and foreshadows a promising integration of computational sciences in understanding complex physical behaviors.